I agree with Grassbank and Pete S: G&N have traditionally been our biggest rivals - partly for their geographical proximity. They ceased to be our rivals in our wilderness years, when we dropped back to the Kent League while they were moving up divisions to the Conference. So it's been a long time since we've actually played them for the rivalry to be expressed (for all the stattos out there: when was the last time we played them competitively? I would guess in the Kent Senior Cup sometime).
Our on-the-pitch rivalry with Maidstone doesn't date back so far and was, I suggest, sporadic at best since we weren't in the same league all that often. In the last 15-16 years the rivalry has been almost entirely off-the-pitch, stoked up by the ground-share and both clubs going bust, with DFC picking up the tag. It's taken 14 years since then to get the rivalry back on the pitch.
I wouldn't say xdover have ever been real "rivals" as such until a couple of years ago, started off by what might be called that rather "tasty" FA Trophy match and replay when the Darts beat the higher-league xdover. Until then, they had just represented another, rather farther-away-than-most "local" derby in the few seasons we were in the same league.